The end goal is to find the set of values of $f$ and $N$ for which
the system has solutions for which $0 \lt 1 - f - \alpha_i \lt 1$ for all the range of $N \gt K \gt 0$, which (i think) is equivalent to asking that $\Pi_{i=j}^{K}(1-f- \alpha_i) \lt 1$. I refer to this quantity as the remainder flux, because it can be interpreted as the net coefficient of attenuation after each node consumes a fixed ratio $f$ and a variable ratio $\alpha_i$

Doing some numerical experiments i've found that for $N=4000$ , $f=10^{-3}$, the system for $K=18$ is the highest for which there are solutions that satisfy the condition, but for $K=19$ the remainder flux becomes above one for $j=1$. On the other hand, for $f=10^{-4}$, the system has solutions for all $K \lt 4000$

I've been looking into how to figure how to find what values of $f$ and $N$ will produce "stable" solutions. Are there any suggestions that can help me?

If I use the previous equation,
and assume that $f$ is small,
it becomes
$(1-f)^N(1-(1-f)^N) \approx 2f^2$.

If $f$ is small compared to $1/N$,
which may not be true by your computation,
this becomes
$(1-Nf)(Nf) \approx 2f^2$
or
$1-Nf \approx 2f/N$
or $f \approx 1/(N+2/N) \approx 1/N$.
For $N=4000$,
this gets $f \approx 2.5\times 10^{-4}$,
which is not too far from your computation.

As a guess, I would say that
$f$ has an expansion of the form
$a/N + b/N^2$ for
$a$ close to $1$ and some $b$.